No, you won't be safe, but you won't accidentally log into your gmail account from Linkedin. This is specifically about Linkedin, because it's asking for your gmail credentials and if in a hurry, as I often am and don't go to linkedin that often, I mistakenly enter my linkedin password thinking it's telling me I am not logged in.
I agree that letting linkedin in scan your 11,000 contacts is going to be faster, but seriously, how many contacts are useful? I have less than 100, and some of those are kind of iffy. Unless you are a recruiter that has a use for perusing random resumes, it seems unlikely to get much value from having 10k profiles linked.
I think some people see it as a measure of their value as a professional. If you have 500+ connections, then you've obviously been around and know everyone. This impression can increase your commercial value when you're looking for executive or sales positions. Business is fundamentally social, and if you can leverage a large network in favor of your employer, you become much more valuable than a smooth talker with no connections.
It does kind of ruin the "introduce me" featureset when everyone you contact is going to say "Actually, err, I can't introduce you guys because I don't actually know him", though.
LinkedIn should mitigate this by minimizing or not even displaying the "number of connections" on profiles and emphasizing other value measurements that don't cause misaligned incentives.
I agree that letting linkedin in scan your 11,000 contacts is going to be faster, but seriously, how many contacts are useful? I have less than 100, and some of those are kind of iffy. Unless you are a recruiter that has a use for perusing random resumes, it seems unlikely to get much value from having 10k profiles linked.